Highly unlikely it will be possible. The UK doesn't permit 'chain migration'. You could only apply as a blood relative of someone in the UK as far as I understand. You don't say the relation to your UK family so not sure if any of them might give it a try. I am assuming you have no UK born grandparents yourself as this would offer another visa you could qualify for.
You would have very stringent requirements to pass and they are virtually impossible to pass since the UK tightened the rules in 2012. If you choose to apply for this incredibly expensive visa, and it is refused (which is likely), it shuts the door to any future visit visas.
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Also, as bad as things are in SA, it is not one of the countries considered 'high risk' for asylum type of applications.
If you have a directly family member in the UK, you could ask them to relocate to one of the EU countries and try coming through that way (cheaper) but with Brexit looming, this route is not going to be possible for long and is under heavy scrutiny already.